Smallest \(g\)-supersolution with constraint
type: Item
instance of: scholarly article
title: Smallest \(g\)-supersolution with constraint
zbMATH Open document ID: 0973.60065
published in: Applied Mathematics. Series B (English Edition)
publication date: 2001-11-18T00:00:00Z
review text: This paper studies the backward stochastic differential equation \[ Y_t = \xi + \int_t^T g(s,Y_s,Z_s) ds + A_T-A_t - \int_t^T Z_s dW_s \] with the constraint \(\varphi(t,Y_t,Z_t)\equiv 0\). Assuming the existence of a solution satisfying additional integrability requirements, the author shows the existence of a unique minimal solution within this class. This extends results by \textit{S. Peng} [Probab. Theory Relat. Fields 113, No. 4, 473-499 (1999; Zbl 0953.60059)] to the case where \(g\) and \(\varphi\) are not Lipschitz, but only satisfy a Yamada-type condition; the argument relies on results by \textit{X. Mao} [Stochastic Processes Appl. 58, No. 2, 281-292 (1995; Zbl 0835.60049)]. If \(g^\alpha\) and \(\xi^\alpha\) depend in addition on a parameter \(\alpha\) and the essential suprema of \(g^\alpha\) and \(\xi^\alpha\) over \(\alpha\) again lie in this family, the solution to the problem with the maximal coefficients is shown to be the essential supremum of the solutions \(Y^\alpha\). The key result behind all this is an extension of the comparison theorem for BSDEs to the present non-Lipschitz case.
reviewed by: Martin Schweizer
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 91G80
Mathematics Subject Classification ID: 60G48
zbMATH DE Number: 1542570
zbMATH Keywords: backward stochastic differential equations
zbMATH Keywords: constraints
zbMATH Keywords: comparison theorem
zbMATH Keywords: Lipschitz condition
author: Qingquan Lin
MaRDI profile type: Publication
full work available at URL: s11766-000-0053-0
OpenAlex ID: W84266789
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DOI: 10.1007/S11766-000-0053-0
description: scientific article; zbMATH DE number 1542570